Expression of type ExprTuple

from the theory of proveit.physics.quantum.algebra

import proveit
# Automation is not needed when building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_expr # Load the stored expression as 'stored_expr'
# import Expression classes needed to build the expression
from proveit import Conditional, ExprTuple, Lambda, U, i, n
from proveit.core_expr_types import v_1_to_n, v_i, w_1_to_n, w_i
from proveit.linear_algebra import OrthoNormBases, Unitary, VecSum
from proveit.logic import CartExp, Equals, Exists, Forall, InSet, Set
from proveit.numbers import Complex, Interval, NaturalPos, one
from proveit.physics.quantum import Bra, Ket, Qmult
from proveit.physics.quantum.algebra import v_1_to_n_kets, w_1_to_n_kets

# build up the expression from sub-expressions
sub_expr1 = OrthoNormBases(CartExp(Complex, n))
expr = ExprTuple(Lambda(n, Conditional(Forall(instance_param_or_params = [U], instance_expr = Exists(instance_param_or_params = [v_1_to_n, w_1_to_n], instance_expr = Equals(U, VecSum(index_or_indices = [i], summand = Qmult(Ket(v_i), Bra(w_i)), domain = Interval(one, n))), conditions = [InSet(Set(v_1_to_n_kets), sub_expr1), InSet(Set(w_1_to_n_kets), sub_expr1)]).with_wrapping(), domain = Unitary(n)), InSet(n, NaturalPos))))

expr:

# check that the built expression is the same as the stored expression
assert expr == stored_expr
assert expr._style_id == stored_expr._style_id
print("Passed sanity check: expr matches stored_expr")

Passed sanity check: expr matches stored_expr

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\left(n \mapsto \left\{\forall_{U \in \textrm{U}\left(n\right)}~\left[\begin{array}{l}\exists_{v_{1}, v_{2}, \ldots, v_{n}, w_{1}, w_{2}, \ldots, w_{n}~|~\left\{\left\{\lvert v_{1} \rangle, \lvert v_{2} \rangle, \ldots, \lvert v_{n} \rangle\right\}\right\} \in \textrm{O.N.Bases}\left(\mathbb{C}^{n}\right), \left\{\left\{\lvert w_{1} \rangle, \lvert w_{2} \rangle, \ldots, \lvert w_{n} \rangle\right\}\right\} \in \textrm{O.N.Bases}\left(\mathbb{C}^{n}\right)}~\\
\left(U = \left(\sum_{i=1}^{n} \left(\lvert v_{i} \rangle \thinspace \langle w_{i} \rvert\right)\right)\right)\end{array}\right] \textrm{ if } n \in \mathbb{N}^+\right..\right)

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	ExprTuple	1
1	Lambda	parameter: 80 body: 2
2	Conditional	value: 3 condition: 4
3	Operation	operator: 5 operand: 8
4	Operation	operator: 57 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	80, 9
8	Lambda	parameter: 33 body: 11
9	Literal
10	ExprTuple	33
11	Conditional	value: 12 condition: 13
12	Operation	operator: 14 operand: 17
13	Operation	operator: 57 operands: 16
14	Literal
15	ExprTuple	17
16	ExprTuple	33, 18
17	Lambda	parameters: 19 body: 20
18	Operation	operator: 21 operand: 80
19	ExprTuple	23, 24
20	Conditional	value: 25 condition: 26
21	Literal
22	ExprTuple	80
23	ExprRange	lambda_map: 27 start_index: 79 end_index: 80
24	ExprRange	lambda_map: 28 start_index: 79 end_index: 80
25	Operation	operator: 29 operands: 30
26	Operation	operator: 31 operands: 32
27	Lambda	parameter: 93 body: 88
28	Lambda	parameter: 93 body: 89
29	Literal
30	ExprTuple	33, 34
31	Literal
32	ExprTuple	35, 36
33	Variable
34	Operation	operator: 37 operand: 41
35	Operation	operator: 57 operands: 39
36	Operation	operator: 57 operands: 40
37	Literal
38	ExprTuple	41
39	ExprTuple	42, 44
40	ExprTuple	43, 44
41	Lambda	parameter: 87 body: 45
42	Operation	operator: 60 operand: 52
43	Operation	operator: 60 operand: 53
44	Operation	operator: 48 operand: 54
45	Conditional	value: 50 condition: 51
46	ExprTuple	52
47	ExprTuple	53
48	Literal
49	ExprTuple	54
50	Operation	operator: 55 operands: 56
51	Operation	operator: 57 operands: 58
52	Operation	operator: 60 operands: 59
53	Operation	operator: 60 operands: 61
54	Operation	operator: 62 operands: 63
55	Literal
56	ExprTuple	64, 65
57	Literal
58	ExprTuple	87, 66
59	ExprTuple	67
60	Literal
61	ExprTuple	68
62	Literal
63	ExprTuple	69, 80
64	Operation	operator: 85 operand: 77
65	Operation	operator: 71 operand: 78
66	Operation	operator: 73 operands: 74
67	ExprRange	lambda_map: 75 start_index: 79 end_index: 80
68	ExprRange	lambda_map: 76 start_index: 79 end_index: 80
69	Literal
70	ExprTuple	77
71	Literal
72	ExprTuple	78
73	Literal
74	ExprTuple	79, 80
75	Lambda	parameter: 93 body: 81
76	Lambda	parameter: 93 body: 82
77	IndexedVar	variable: 90 index: 87
78	IndexedVar	variable: 91 index: 87
79	Literal
80	Variable
81	Operation	operator: 85 operand: 88
82	Operation	operator: 85 operand: 89
83	ExprTuple	87
84	ExprTuple	88
85	Literal
86	ExprTuple	89
87	Variable
88	IndexedVar	variable: 90 index: 93
89	IndexedVar	variable: 91 index: 93
90	Variable
91	Variable
92	ExprTuple	93
93	Variable